↑

Properties

Label 18.10.1981841248...1856.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{18}\cdot 3^{32}\cdot 19^{3}\cdot 29^{6}$
Root discriminant $70.77$
Ramified primes $2, 3, 19, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T366

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6859, 0, -35739, 0, 27360, 0, 161004, 0, -81936, 0, 117, 0, 2481, 0, -99, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 99*x^14 + 2481*x^12 + 117*x^10 - 81936*x^8 + 161004*x^6 + 27360*x^4 - 35739*x^2 - 6859)
 
gp: K = bnfinit(x^18 - 18*x^16 - 99*x^14 + 2481*x^12 + 117*x^10 - 81936*x^8 + 161004*x^6 + 27360*x^4 - 35739*x^2 - 6859, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 99 x^{14} + 2481 x^{12} + 117 x^{10} - 81936 x^{8} + 161004 x^{6} + 27360 x^{4} - 35739 x^{2} - 6859 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1981841248485867753613659602681856=2^{18}\cdot 3^{32}\cdot 19^{3}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.77$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 19, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9} a^{6} + \frac{1}{3} a^{4} + \frac{2}{9}$, $\frac{1}{9} a^{7} + \frac{1}{3} a^{5} + \frac{2}{9} a$, $\frac{1}{9} a^{8} + \frac{2}{9} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{9} + \frac{2}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{2}{9} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{81} a^{12} - \frac{1}{27} a^{10} + \frac{4}{81} a^{6} - \frac{2}{27} a^{4} + \frac{4}{9} a^{2} - \frac{23}{81}$, $\frac{1}{81} a^{13} - \frac{1}{27} a^{11} + \frac{4}{81} a^{7} - \frac{2}{27} a^{5} + \frac{4}{9} a^{3} - \frac{23}{81} a$, $\frac{1}{12312} a^{14} + \frac{5}{3078} a^{12} + \frac{25}{1026} a^{10} - \frac{635}{12312} a^{8} + \frac{383}{12312} a^{6} + \frac{1967}{4104} a^{4} - \frac{3821}{12312} a^{2} - \frac{25}{648}$, $\frac{1}{12312} a^{15} + \frac{5}{3078} a^{13} + \frac{25}{1026} a^{11} - \frac{635}{12312} a^{9} + \frac{383}{12312} a^{7} + \frac{1967}{4104} a^{5} - \frac{3821}{12312} a^{3} - \frac{25}{648} a$, $\frac{1}{142661213499456} a^{16} + \frac{3764240177}{142661213499456} a^{14} - \frac{2121653933}{990702871524} a^{12} - \frac{972762673919}{142661213499456} a^{10} - \frac{135368273033}{17832651687432} a^{8} + \frac{264275835905}{5944217229144} a^{6} + \frac{6876135972253}{35665303374864} a^{4} - \frac{37884502603}{1877121230256} a^{2} + \frac{34452655547}{131727805632}$, $\frac{1}{142661213499456} a^{17} + \frac{3764240177}{142661213499456} a^{15} - \frac{2121653933}{990702871524} a^{13} - \frac{972762673919}{142661213499456} a^{11} - \frac{135368273033}{17832651687432} a^{9} + \frac{264275835905}{5944217229144} a^{7} + \frac{6876135972253}{35665303374864} a^{5} - \frac{37884502603}{1877121230256} a^{3} + \frac{34452655547}{131727805632} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 27079382469.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T366:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n366
Character table for t18n366 is not computed

Intermediate fields

3.3.2349.1, \(\Q(\zeta_{9})^+\), 9.9.1049866478469.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.12.4$x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 38 x^{2} - 31$$2$$6$$12$12T87$[2, 2, 2, 2, 2]^{6}$
$3$3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
29.12.6.1$x^{12} + 146334 x^{6} - 20511149 x^{2} + 5353409889$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$